1. Field of the Invention
A method for economically de-aliasing spatially-aliased seismic data using trace interpolation methods confined to the f-k domain.
2. Discussion of Relevant Art
The art of seismic exploration requires that a plurality of acoustic receivers be disposed at designated stations distributed at regular spatial intervals over a line or area of survey. An acoustic source visits selected stations over the survey area whereupon at each station visited, it radiates an acoustic wavefield into the earth. The wavefield propagates in all directions, insonifying sub-surface strata. The wavefield is reflected from each stratum in turn, to return to the surface where the reflected wavefield is detected by the receivers.
The acoustic receivers convert the mechanical motions and pressure variations due to the propagating wavefield to electrical analog signals. The electrical signals are sent to suitable recording and processing equipment over data transmission channels of any desired type. The analog signals representative of the received acoustic wavefields are recorded as a periodic function of signal level vs. two-way wavefield reflection travel time. The periodic electrical analog signals, as seen by each of the receivers, are discretized in the time domain at selected sample intervals and then recorded as a plurality of discrete time-scale traces, one trace for each station occupied by a receiver. A suite of traces recorded at a single receiver station, due to wavefields generated from a set of sequentially-visited source locations, might comprise a common receiver gather in the time-space (t-x) domain.
The useful temporal frequency of the received seismic data lies in the range of 5 to about 125+ hertz. In data-sampling theory, it is fundamental that no wavelength embedded in the data can be shorter than twice the sampling interval, otherwise the data are distorted due to aliasing. Given a selected temporal sample interval, the received seismic signals may be electrically or optically low-pass filtered before digitization in the time domain to exclude frequencies above the aliasing frequency, which is also referred to as the temporal Nyquist or temporal folding frequency. Thus for a 4-millisecond (ms) sample interval, frequencies above 125 Hz (.lambda.=8 ms) must be excluded. Insofar as a single one-dimensional time-scale trace is concerned, selection of the proper sample interval and the corresponding anti-aliasing filter is a simple operator-selectable instrumental option.
In real life, seismic data are multi-dimensional rather than uni-dimensional. Along a preselected line of survey, including a plurality of serially adjacent seismic traces the data reside in a 2-D, temporal-spatial (t-x) domain. In an areal survey, the data may be in the 3-D (t-x-y) domain. In the presence of steeply-dipping strata, if the source or detector station spacing is too wide relative to the spatial frequency of a dipping stratum, the data will be spatially aliased.
FIG. 1A is a set of 24 synthetic time scale traces plotted from t=0.0 millisecond (ms) to t=500.0 ms as derived from a number of common midpoint (CMP) gathers. Three parallel events are shown having a spatial dip of 8 ms per trace in the positive direction as shown by sloping line 10. Because of the wide spacing between traces, it would be easy to mistakenly draw false or aliased negative dips of about 2.5 ms per trace as shown by sloping line 12, particularly if the data had been partially obscured by random noise. If now, there had been available a set of traces having half the spacing of FIG. 1A, such as in FIG. 2A where the dip is but 4 ms per trace, there would be little doubt as to the interpretation of the correct dip.
By definition, the spatial phase shift of a coherent event is the time difference between the same event as seen on two adjacent traces. Given a cluster of events having reasonably uniform dips characterized by a narrow range of spatial frequencies, then it would be a simple matter to average the phase shift between traces and interleave the averaged traces between the original traces of FIG. 1A as suggested by FIG. 2A. But real data includes a whole spectrum of spatial frequencies and phase shifts. Therefore such a simplistic approach is not possible.
As is well known to the art, the spectrum of spatial frequencies is best examined by transforming the time scale data from the t-x (time-space) domain to the f-k (frequency-wavenumber) domain using a fast Fourier transform by way of example but not by way of limitation (the wavenumber is the number of cyclic periods per unit distance). FIG. 1B is the 2-D Fourier-transformed amplitude spectrum of FIG. 1A. The data are aliased (folded) at 62.5 hertz (Hz) where the events are wrapped around the central or zero ordinate of the wavenumber axis at an angular phase shift of .pi..
As a matter of definitions, the temporal alias frequency in the f-k domain is -1/2.DELTA.t&lt;f&lt;+1/2.DELTA.t. In the spatial coordinate, the spatial alias frequency or wavenumber is -.pi./.DELTA.x&lt;k&lt;+.pi./.DELTA.x. Here .DELTA.t is the sample-time interval in milliseconds and .DELTA.x is the station spacing in preferred units.
As previously stated, selection of the temporal sample interval is a simple user-selectable, instrumental option. But selection of the spatial sampling interval on the ground is much more complex. The station spacing needs to be as sparse as possible for economic reasons because operating costs per seismic station are very expensive. The problem becomes exacerbated in designing a survey for a previously unexplored area having unknown dips. If the spatial frequencies in the area on the average, turn out to exceed the Nyquist limit expected when the original survey strategy was laid out, it may not be economical or, perhaps not even possible such as because of political upheaval, to return to the region to physically acquire additional coverage. Aliased spatial frequencies wreak havoc with migration and interpretation of the seismic data sets as explained earlier.
An interpolation scheme in the f-x domain has been developed by S. Spitz in a paper entitled Seismic Trace Interpolation in the F-X Domain, Geophysics, v. 56, n. 6, June, 1991, pp 785-794. The method is based on the fact that linear events present on a section made of equally spaced traces may be interpolated exactly, regardless of the original spatial interval, without any attempt to determine their true dips. The predictability of linear events in the f-x domain allows the missing traces to be expressed as the output of a linear system, the input of which consists of the recorded traces. The interpolation operator is obtained by solving a set of linear equations whose coefficients depend only on the spectrum of the spatial prediction error filter defined by the recorded traces. The prediction error filter is obtained from the known data at half the temporal frequency. The Spitz method involves two sets of linear equations, one for the prediction error filter and one for the unknown data. Accurate calculation of the unknown data requires edge-free design of the equations. This condition leads to a non-Toeplitz matrix structure that makes dataprocessing by the Spitz method very computer-intensive and therefore very costly.
Another method is disclosed in UK patent application GB-2,282,665-A in the name of Helmut Jakubowicz, published 12/04/95. In this disclosure, each data set such as a common receiver gather is applied to a sinc filter to generate additional data for interpolation intermediate the recorded actual data. The recorded actual data are also subject to traveltime angularity correction. Interlacing sinc data interpolated with the known data does not unwrap the spectrum but produces a spatially band-limited spectrum. Furthermore, temporal frequencies above a certain critical frequency remain wrapped along the k-axis which might necessitate temporal band-limiting of the data.
There is a need for an economical computer-implemented method for interpolating spatially aliased, multi-dimensional data sets.
SUMMARY OF THE INVENTION
A method for unwrapping spatially aliased signals representative of an acoustic wavefield propagated from an acoustic source. The signals representative of the wavefield are sampled at each of a first plurality of signal-sampling locations which are spaced apart by known distances. The signal samples from the respective sampling locations are formatted as a recorded gather of sequentially-ordered time-scale traces. The time-scale traces of the recorded gather are transformed from the t-x domain to the frequency-wavenumber (f-k) domain to form a transformed gather, the temporal frequency range f is (0, F.sub.N) and the range of wavenumbers k is (-K.sub.N, +K.sub.N) where F.sub.N and K.sub.N are the temporal and spatial Nyquist frequencies respectively. A first transform gather is formed from the odd numbered traces of the recorded transform gather for temporal frequencies between zero and F.sub.N /2 and for spatial frequencies between -K.sub.N /2 and +K.sub.N /2. A second transform gather is similarly formed from the even numbered traces of the recorded transform gather for the corresponding temporal and spatial frequencies. The temporal frequencies of the first and second transform gathers are stretched from range (0, F.sub.N /2) to range (0, F.sub.N) by interpolation thereby to form third and fourth transform gathers. Next, the spatial frequencies of the third and fourth transform gathers are stretched from range (-K.sub.N /2, +K.sub.N /2) to range (-K.sub.n, +K.sub.N) to form fifth and sixth transform gathers. The sixth transform gather is divided by the fifth transform gather to define the f-k transform of an interpolation operator U. The f-k transform of the interpolation operator U is multiplied point-by-point with the original f-k transform of the recorded gather to form a complex product. The complex product is inverse transformed from the f-k domain to the t-x domain to provide a second gather of time scale traces. The respective traces of second gather of time-scale traces are interleaved between the time-scale traces of the original recorded gather.